Friday, August 10, 2018

Young Children and Subtraction


This post is a continuation of my post on teaching math to young children here.

Chapter 6 of Kamii’s book (Young Children Reinvent Arithmetic) talks about the difficulty young children have with subtraction.  She points out that many math textbooks introduce subtraction shortly after the introduction of addition to 5 or 6-year-olds, however subtraction involves thinking negatively about quantities—or thinking about the absence of a quantity, which is difficult for children at this age.  Experiments that she describes in the chapter show that until age 7 many children have a difficult time thinking about amounts negatively, and consequently subtraction is a pretty complicated operation for them.  The exception, to this rule, is in word problems.

When children are given word problems, they often will use addition to solve a subtraction problem in their head.  Word problems give more meaning to the concept of subtraction and so many 5 and 6-year-olds can figure them out without instruction—and even those who don’t will benefit from the mental process of trying occasionally.

Kamii also points out that as children become very good at addition and are able to immediately give answers to addition problems such as 5 + 5 or 6 + 1, they become able to easily give the answers to the corresponding subtraction problems (10 – 5 and 7 – 1).  So rehearsing addition problems through games seems to be a good way to build in children an ability to subtract mentally as well.

My take-ways from all of this:
For Finn’s Kindergarten and First grade years, II will focus on the teaching of addition through games and present him with word problems that represent both addition and subtraction, but I will not expect him to solve written subtraction problems yet.

Wednesday, August 8, 2018

A Different Perspective on Teaching Math to Young Children


During my graduate studies in elementary education at the University of Alabama at Birmingham, I was privileged to take a course on teaching elementary math from Constance Kamii, a professor who had worked closely with and learned much from Jean Piaget, a Swiss psychologist known for his work in childhood development.  This course completely changed my understanding of how children attain mathematical concepts and made it abundantly clear to me why so many people believe themselves to be hopelessly bad at math.

The course centered around a book written by Constance Kamii, “Young Children Reinvent Arithmetic,” and I have since reread and reread this book as I seek to apply what I learned with those who I teach.  Now, as my oldest is approaching Kindergarten in the next year or so, I am rereading the book again with pen in hand in order to arrive at my plan for introducing Finn to math.

One of the simplest conclusions based on all of this study that I can share with you, is that the way traditional textbooks and teachers introduce children to numbers and algorithms (processes for arriving at correct answers to math problems) often derails them in their ability to think for themselves mathematically.  Children who would be able to, in time, do large sums in their heads with a full understanding of what they are doing, are instead directed to follow step 1, then step 2, then step 3 in order to produce an answer and they have no real understanding of why they are doing said steps or why the answer they arrived at is correct (or not correct).

So, for starters, I plan to avoid introducing algorithms to Finn as long as possible and allow him to grow in his concept of numbers and how they relate to each other through playing lots of math games that encourage mental math.  In Kamii’s book there are countless studies and experiments conveyed with school students which show that learning math in this way early on leads to correct answers a much higher percentage of the time, a greater understanding of how to solve real-world math problems, and a greater joy and confidence for children in doing math.

I plan to continue journaling my findings and subsequent plans as I go through Kamii’s book yet again.

Tuesday, August 7, 2018

My Study of Exodus



I love the imagery being presented in the Old Testament—imagery that points to what God was planning to do much later. A couple years ago I started studying the book of Exodus and those New Testament/present day connections that can be seen.  It was a very enlightening study that enhanced my understanding of the gospel and the holistic meta narrative of the Bible.  I subsequently began a similar study of Leviticus and am currently about 2/3 of the way through that book.  Here are my notes on Exodus.  *Please excuse the formatting issues and don't try to look at this on a mobile device (apparently posting a table on a webpage is complicated!)

Exodus
God’s children, the Israelites, were in slavery to the Egyptians
Ex 1:11, Ex 2:23, Ex 3:7
We, God’s children, are born into slavery to sin.
John 8:34, Romans 6:6, 16-17, & 20, Romans 7:14
God beckons His people to leave Egypt and come worship Him.
Ex 3:7-10, 16-17, Ex 4:22-23, Ex 6:6-8
God beckons us to leave our sin and live lives of worship to Him.
John 4:23-24, Romans 12:1-2, Hebrews 12:28-29
Pharaoh refused to relinquish his grip on God’s people.
Ex 5:2
Sin refuses to relinquish its grip on us.  Try as we might, we cannot get free of it on our own.
Romans 7:14, 24, Romans 8:7-8
God sends the plagues on Egypt, culminating in the visitation by angel of death on every first born Egyptian.  Ex 12:29-30
God declares that death is the just punishment for all who sin.  Genesis 2:16-17, Romans 6:23
The Israelites, however are provided a way to avoid this death through observing the first Passover.  A perfect lamb is sacrificed and the sign of its blood keeps the angel of death away from their household.
Ex 12:1-30
Jesus Christ is our Passover lamb. Through his death and shedding of blood, we are provided a way of escape from the just judgement for our sin. John 1:29, 1 Peter 1:18-20, Ephesians 1:7, Colossians 1:20, Hebrews 10:5-10
The Israelites were instructed to choose a year-old lamb without blemish, take it into their homes, and care for it for 4 days before slaughtering them together as a community.  Ex 12:5-6
Jesus came to the people as a sinless human, lived for 33 years among them, and then they crucified him.

Also, it was on the day that lambs were chosen for the Passover that Jesus entered Jerusalem on a donkey.  God, in a sense, was presenting His chosen spotless lamb, just as all Israelites were choosing theirs.  Jesus was celebrated, and lived in Jerusalem with his people for 4 days.  Then they crucified him as a community.

Mark 11, John 12
The Israelites were instructed to eat the meat of the Passover lamb along with unleavened bread and bitter herbs.  Exodus 12:8
At the last supper Jesus shared with His disciples, He instructed them to consider the bread his body, and the wine, his blood.  
Matthew 26:26-28, John 6:53-57
No bone of the lamb was to be broken.
Ex 12:46, Numbers 9:12
Not one of Jesus’ bones was broken
John 19:33
Immediately after observing the Passover, the Israelites were to observe the 7-day long Festival of Unleavened Bread.  7 days, that is, of abstaining from consuming any leaven.  It was to be completely removed from their houses, and anyone who did eat any was to be cut off from the nation of Israel.
Ex 12:14-20
The timing of this festival is interesting to me because leaven so often represents sin in the Bible (Matt. 16:5-12, 1 Cor. 5:6-8), and it was unleavened bread that Christ held up at the last supper declaring it to be His body (which was without sin).  Just as the Israelites were called to give up leaven after their salvation from Egypt had been secured, we are called to leave behind our sins as a response to the salvation we have in Christ.
The Israelites were instructed to teach their children regarding the festival of unleavened bread that it was because of how God saved them from Egypt, that they observe these seven days without leaven.
Ex 13:7-8
As believers we are taught to leave behind sin as a response to the salvation and freedom from slavery to sin that we have in Christ.
Romans 6, 1 Cor. 5:6-8, Hebrews 12:1, 1 Peter 2:24, 1 John 3:9 and 5:18
The most direct route to the promised land God had for His children would have meant an immediate battle with the Philistines, and God realized that His people were not ready for that.  They would cower and return to Egypt.  So instead of taking them that way, he leads them into the wilderness, where they would follow Him for 40 years.
Ex 13:17:18
I think that just as God had definitive and good plans for the people of Israel, He has definitive and good plans for us, but He knows us, and knows the measure of faith He has bestowed upon us, and so sometimes His best for us is not the quickest route to a destination.  Sometimes He leads us into the wilderness for many years to grow our faith and prepare us for the good things He has for us later on.
Romans 12:3, Psalm 23, 2 Corinthians 12:9-10, Romans 5:3-5, James 1:2-4, 1 Peter 1:6-7
God leads His people into a situation where they see the Egyptians coming after them and then miraculously saves them from the Egyptians through opening up the sea for them and then closing it again upon Pharaoh’s army.
Ex 14
Sometimes as believers we become overwhelmed by our sin and cry out as helpless children, unable to conquer it on our own.  We become desperate, and it is in that posture that God chooses to show us deliverance.  Just as the Israelites were completely unable to escape the Egyptians on their own, we are completely unable to escape our slaver to sin on our own.  Sometimes we try, and our trying proves useless.  God alone could defeat the Egyptians for the Israelites, and God alone can bring us freedom from our sin when we throw up our hands and surrender to His plan.  1 Peter 5:6-10
After God delivered the Israelites from the Egyptians, the people rejoice and even write songs to commemorate what the Lord did.
Ex. 15:1-22
I think this is a great example of what we OUGHT to do when God has done something amazing for us.  We should commemorate it in some way and make a big deal out of it.
When the people could not find water, they grumbled against Moses, their God-appointed leader.  Then God miraculously provided for them through Moses’ throwing a stick into a well.  God also chose to give them a special promise, conditional on their obedience.  Then God brought them to a lush oasis in the wilderness.
Ex 15:22-27
God sometimes leads us into situations that don’t seem best to us, and works through people who don’t seem wise or capable to us.  Often there are beautiful oasis’s not very far away from us, but we can’t see them yet, and disbelieve the goodness of God’s plan for us.
The Israelites complain about their circumstances again and wish that they could be back in Egypt
Ex 16:1-3
This desire to be back in Egypt might seem crazy to us since we know they were slaves in Egypt, but so often we see the lives of non-Christians—people still in slavery to sin, and wish that we could “enjoy” the fun that they do.
The Lord provides Manna for the people, and instructs them to take only what they need for each day and double on the 6th day so that they can rest on the 7th.
Ex 16:4-31
God always provides for His people - even through unusual means sometimes.  He also demands that we rest, and if we do not obey, He often forces it on us like He did with the Israelites.  The choice to rest is an act of faith.
God commanded the people to keep some of the manna along with other mementos of His goodness for generations to come.
Ex 16:32-36
We could keep mementos of God’s goodness to us for generations to come as well.
Ex 17 The people complain, and God shows up with two miracles - one by providing water from a rock, and second by giving them a miraculous defeat of their enemies, the Amalekites.
God often comes through for us when we feel like we are in dire circumstances and complain.  How much better it would be if we would go to Him in confidence of His good plan, rather than in a spirit of complaint and with the Israelite’s attitude that God didn’t have a good plan.
Ex. 18 God used Moses’ father in law to show him a healthier way of dealing with the conflicts among the Israelites.  Moses had a plan and seemed to be handling things alright, but it must have been exhausting and it wasn’t developing leadership among the people well.
So often I stubbornly handle something myself that could be dealt with a better way if I would humble myself and take advice and help from others.  God often brings others into our lives to lend a hand and help us tackle life in a healthier way.  The question is, will we let them?
Ex 19 - 20

Timeline:
The Israelites celebrated the Passover and then left Egypt at twilight on the 14th of the 1st month of the year (effectually leaving on what would have then been the 15th) (Exodus 12:2, 6, & 51).

Ex 19:1 states that on the 3rd new moon, the people arrived in the wilderness of Sainai.  Because a lunar month is about 30 days long, we can conclude that the people arrived there about 45 days after beginning their journey.

19:2-15 the Israelites camped while Moses went up the mountain to God, came down again with a message for the people, went up to the mountain a 2nd time, and then came down again with instructions to spend 3 days consecrating themselves.  It’s hard to say how many days all of the going up and coming back down took, but it seems clear that by the time we get to verse 16, we are either 49 days, or 7 full weeks, after the Passover, or very close to it.

Lev. 23:15-22; Acts 2 - On or near the day in which the Israelites would celebrate the feast of weeks, or Pentecost, God met the Israelites, called them to be His own special possession, and gave them the 10 commandments, or marital vows to demonstrate their devotion to Him.  On this day throughout the Old Testament, the Israelites were to offer their first fruits of grain to the Lord (the only first fruits offering that included leaven) and sacrifice 7 lambs, a bull, and 2 goats.



In the New Testament, on the day of the Feast of Weeks, God gave the Holy Spirit to his children (Acts 2), and brought in a large “crop” of Jews to the kingdom of God through the preaching of Peter and the other apostles.  

The fact that the grain offering during this festival included leaven is thought to signify the sin that is still present in the children of God until Christ’s second coming.  God called His people to himself both through the Old Testament law and through the gift of the Holy Spirit, and we accept those gifts gratefully though we cannot reciprocate His righteous love fully as we still live in sin.

Just as God brought His children out of slavery and brought them through the dessert of learned dependence to Himself, called them His own treasured possession and gave them a list of rules to live by, or marriage vows to keep, so He calls us to Himself, teaches us dependence on Him, and calls us to obey His word as we follow Him in love.

It’s interesting to me to note that, in Ex. 19, when God came to meet the people, they were not allowed anywhere near the mountain where He would be present because of His Holiness.  However, centuries later on the day of Pentecost, after Jesus has died to give His perfect identity to the children of God, God Himself comes to dwell within each of His people.  What a contrast!  In the Old Testament we see a picture of God’s holiness and inability to allow sin in His presence, and in the New Testament we see how completely Jesus has removed our sin from us in God’s choice to come live within us.

Ex 20 - 31 God begins to give His laws to His people—His conditions for the promise of 19:4-6.

The laws cover how the people should relate to Him, to each other, to their slaves, to their family members, and to outsiders in the land they would come to.  It outlines how to deal with sin and brokenness.  It outlines how to schedule their work week, and when to observe holidays.

He gave them instructions on how to worship Him, how to build a place of worship, how to set up the priesthood and exactly how to rightly come before Him with sacrifices.

An article by Tim Keller about what Old Testament Laws are in effect today:

another article on Thirdmill:
http://thirdmill.org/magazine/article.asp/link/http:%5E%5Ethirdmill.org%5Earticles%5Este_lauer%5Este_lauer.ObeyLaw.html/at/Must%20We%20Obey%20the%20Law?
Ex 21-22 these laws seem to be aimed at protecting slaves, victims of injustice, and those without a natural advocate.
There are countless verses in the Bible about God’s justice for those who are underprivileged in some way.  In fact, it seems that justice, if you read all of the verses which mention it, has more to do with lifting up the broken than with punishing the wicked.  Ez. 16:49-50 suggests that of all of the immoral sin in Sodom, the greatest sin was their pride and consequential neglect of justice to the poor and needy.
Ex 23:1-9 are about personal integrity - a high priority for a child of God, and verses 10-19 are about personal worship.  Many of the things in these verses seem to be a repeat or are repeated at length later, but in this passage, they seem to have a more personal nature aiming at the heart of the individual.  God wants to be first in everything and to have the complete trust of his children.  Trust for provision of our physical needs and also trust for justice in our lives and the lives of those around us.
These verses seem to speak to sins that we often justify in particular instances.  Respectable sins, we might call them, because of our circumstances, or because of the circumstances of others involved.

God has something much better than compromise for His children.

In 23:25-33 God gives further blessings that will come with obedience to his laws and His leading regarding their conquest of the promise land.

Ex 26-27 God describes in great detail exactly how He wanted His tabernacle to be designed.  It was to be very ornate and beautiful in some senses, but also easily packed up and moved (it was a tent), so in some senses it was very modest.
The tabernacle of the Old Testament foreshadowed the temple that Solomon would construct for the Lord, which itself foreshadowed the New Testament temples - believers in Christ (1Corinthians 3:16).  

In some sense, the differences between the tabernacle of Moses’ day and the temple of Solomon’s day harken to the differences between the way in which we represent the temple of God now, and the way we will do so more perfectly one day, when sin and death are no more.  God’s presence dwelled in the tabernacle just as truly as he did in Solomon’s temple, and the tabernacle was still a truly sacred and holy place, but it was still a tent.  It still resided in a desert wilderness and had to be taken apart and reconstructed on a regular basis.  We, like the tabernacle, house the Spirit of God in a modest tent.  We live in a dessert region and we are regularly undone by life’s circumstances.  Yet one day we will possess a more permanent and perfect body of splendor which reflects more perfectly the holiness which has been bestowed upon us.
Ex 28-29, and 39 describes the priest’s garments and the consecration of the priests.  The consecration of the priests is described in more detail in Leviticus 8.
The description of the garments worn by the priests speaks to us of the holiness of our spiritual clothing in Christ as priests of the new covenant.  The priests were consecrated with water and with oil.  Just as sacrifices were made to consecrate the priests for service to the Lord, Jesus’ death consecrates us for this holy service (Hebrews 9:11-14)
In 31:1-11 God tells Moses that He has uniquely equipped certain men with artistic ability to create the items and designs He wanted in His tabernacle.

Ex 32-34 While Moses is receiving the ten commandments from God, Aaron leads the people in creating the golden calf.  After seeing their sin Moses breaks the two tablets and God declares that He will not travel among the people to the promised land because his anger would “consume them on the way.”  Moses pleads with the Lord on the people’s behalf, declaring that it is God’s presence among them that sets them apart from other people groups.  God answers Moses that He will grant Moses’ request because “you have found favor in my sight, and I know you by name.”  Then Moses asks to see God’s glory and God agrees to let all of his goodness pass before him and proclaim before him His name, “The LORD.”  He said that no man could see His face and live, so he would shield Moses from His face while He passed before him, but then let him see His back.

Moses spends 40 days and 40 nights with the Lord on Mount Sainai without food or water and cuts two more tablets of stone like the first ones with the ten commandments.  God renews his covenant with the people.

When Moses returns from the mountain his face is shining because he has spoken with God.
Interesting notes:
·                Aaron, who had seen first-hand the power and providence of God, commits blatant idolatry and leads others in it, and yet after this incident God still makes him high priest in the tabernacle.  If God did this, then surely anyone with any past can not only be a Christian, but a leader in the church as well.
·                God chooses to be inclined toward mercy and grace as a result of Moses’ pleading and arguing with Him.  Moses clearly has a very special relationship with God.  I wonder if this kind of relationship possible for you or me today?
·                It’s interesting to me that Moses’ interaction with God this time, after the grave sin of the golden calf (and Moses’ hot-headed breaking of the stone tablets), seems to be more intimate than his interactions with God before - even though the past interactions included the giving of the original commandments and laws, and the giving of the covenant promise to make Israel His people.  Perhaps the grace afforded by the people’s sin somehow enabled a deeper relationship for God and His people than what was possible before.  Why was it only now that Moses requested to see God’s glory?  Why did Moses’ face not shine after earlier encounters with God?  Perhaps the realization of Israel’s sin gave him the boldness to “put it all on the line.”  There was not more pretense of deserving anything from God.  Moses now knew He had to either “go big or go home,” because those were his only options.  And as a result, God chose to lavish His grace on Moses.  Perhaps this is why God sometimes allows His children to fall very far into sin—for the sake of the intimacy with Him that can follow.
Ex 35 - 36 When Moses assembles the people together after their sin and his mediation with God on their behalf, the first thing He relays from God is instructions on observing the Sabbath.

Second, he commands that anyone with a “generous heart” contribute the supplies, time, and talent needed to construct the Tabernacle.

It seems that the Spirit moved many people to give generously.
It’s interesting to me that instead of going on about idol worship and the detestableness of their sin with the golden calf, God instead has Moses instruct the people on proper worship - worship that involves rest from labor as a pivotal aspect.

God could have demanded the supplies necessary for the tabernacle from His people, but instead He chose to encourage giving from those whose hearts moved them (and He certainly was involved in moving people’s hearts this way).
Ex 36:1-7 and 39:21-31 The people bring free-will offerings to the craftsmen for the construction of the tabernacle.  They continue bringing so much, in fact, that they have to tell the people to stop.  The tabernacle would be constructed in its entirety from freewill offerings by the people.
1 Cor. 3:16-17, 6:19, and 1 Peter 2:5 all speak of us today as embodying the temple of God.  We are the present-day tabernacle, housing the Spirit of God and offering spiritual sacrifices.

I think to some extent, however, our bodies can function as temples only to the extent that we offer it to the Lord for that purpose.  Just as God asked the people to give toward the construction of the tabernacle, He asks us what sacrifice we will make, what will we give toward the construction of His tabernacle within us?
Ex 36:8- Construction of the coverings for the tabernacle.  The tabernacle had 4 curtain coverings, each slightly larger than the one under it, so that from the outside, only the outermost curtain would be exposed and visible:

vs. 36:8-9 and 26:1-6 The innermost covering was of fine, colorful linen with woven cherubim. This curtain would be clasped to the second curtain with gold clasps.

vs. 36:14-18 and 26:7-13 the second covering was made of goat hair

vs. 36:19 and 26:14 the 3rd covering was of tanned ram skin, and the final covering was of goat skin or other durable leather.

The veil covering the doorways to the Holy Place and Most Holy Place were made of this colorful fine linen as well, and when the people moved from one location to another, this veil from the doorway was laid over the ark to cover it as they traveled (Numbers 4:5).

The outer covering of goat’s skin would have been better suited to withstand the elements and it was very plain.  Just as to the world’s eyes, we are nothing special.  There is nothing glorious about our appearance. 

The ram’s skin, which lied beneath the goatskin was died red (according to many translations) which can symbolize the blood of Christ which covers us.  Also, a ram was the animal God chose as a substitute for Isaac, symbolizing Christ’s future sacrifice, and a ram was also sacrificed to consecrate Aaron and his sons for the priesthood.  To the worlds eyes, we may look like goats, but beneath our outer covering is Jesus’ sacrificed body and blood which consecrate us for His service.

This ram’s skin covering would be surrounding the 3rd layer, which was of goat’s hair.  The goat’s hair curtain represents our fallen sinful nature, since goats often represented sinners or waywardness in the Bible.  An example of this was the use of goats on the Day of Atonement (see Leviticus 16).  As plain and scratchy as this curtain would have been, it was covered by the ram’s skin completely around it and covered by the beauty of the linen curtain from within.

From the inside of the tabernacle, the covering which could be seen was that of fine linen, decorated with cherubim.  This represents the way God sees us from within.  We are holy and righteous in His eyes.  There may be a curtain of goat’s hair just on the other side of this fine linen curtain, and we may fail and sin constantly, but God dwells in us nonetheless and sees us as His perfect sanctuary because of the work of Christ on our behalf.

The veil covering the doorways to the Holy Place and Most Holy Place represent the body of Christ.  When Christ died, the veil was torn symbolizing the way He had made for us to approach God.  (Hebrews 10:20)
Ex 37:1-9 Described the Ark of the Covenant.  Also Ex 25:10-22

God would dwell above the “mercy seat,” the golden cover to the ark, and from there He would speak with Moses (Also Numbers 7:89).  

Inside the ark they were to place the tablets with the ten commandments (see Deuteronomy 10:2)

Inside the ark was also placed a golden urn holding manna, and Aaron’s staff that budded (Hebrews 9:4; Ex. 16:33).

Numbers 10:33 when the Israelites traveled, the Ark of the Lord went before them.

Deuteronomy 31:24-25 God had the people place the book of the law beside the Ark as a testimony to them.
Just as God dwelled inside the Holy of Holies in the tabernacle, so He dwells inside of us, His people, His present-day temple.  He leads us from that place, just as He led the Israelites, and He commands us to hold dearly both the law, His words, and the evidences of His work in our life.  The Israelites were to keep the Manna, and Aaron’s budded staff inside the ark, and I think God would have us hold in our hearts memories of the ways He has worked in our lives as well (Deuteronomy 6:20-23, and many other places God has the people set up standing stones as symbols of His work in their lives).
Ex 37:10-16 describes the construction of the table that would hold the bread of the presence.  See Lev. 24:5-9 about the bread of presence.  12 loaves of bread made of fine flour were to be placed on this table at all times, and on the Sabbath, the priests would eat the bread and replace it with 12 new loaves (12 representing the 12 tribes of Israel).  The bread was Holy and only to be eaten by the priests.  It would symbolize how God provides holy and good food for His chosen people.  This table, overlaid with pure gold, would reiterate the holiness and perfection of the food that God provides for His people.
See New Testament verses about God providing holy and life-giving food for His people:
John 6:35
John 6:51-58
John 6:63

I imagine the priests going about their weekly duties in the tabernacle, seeing this table with the loaves positioned on them and being reminded of their high calling as priests and of the privilege to be fed by the hand of God with holy food that commoners were not even permitted to see (as it stayed in the Holy Place).  We, like those priests, have high callings with privileges such as holy spiritual food from the hand of God that non-believers know nothing of.  The question is, do we choose to see it and partake of it ourselves?
Ex 37:17-24 describes the construction of the golden lampstand that would reside in the Holy Place, as the only light-giving object in the tent.  It was to be made of pure gold - carved out of a single gold piece.  It was to have the image of almond blossoms on all of its seven branches.
26:35 instructs that the lampstand should be placed opposite the table (and would shed light on it.)

Leviticus 24:1-4 describes how the people were to bring pure olive oil for the lamp so that the light could keep burning continually.  Aaron, the high priest, was to be in charge of keeping it burning from evening to morning.  It says that this would be “a statute forever throughout your generations.”
There seems to be a lot of symbolism around the golden lampstand.
·                Only pure olive oil would be used to keep it lit.  Olive oil often represented the Holy Spirit.
·                There were seven branches.  Seven is the number of completeness in the Bible.
·                The branches were designed to look like almond tree branches with almond blossoms.  The Hebrew word for the Almond Tree is from the same root as the word for watchfulness, and throughout the Bible the Almond tree is a symbol of God’s watchfulness over His people. (See Jeremiah 1:11-12).
·                It was only the High Priest, who today is Christ, who was to tend the lamp and keep the fire going.

All of this seems to represent the way God promises to keep watch over His people 24-7 and light the way for us with continual guidance.  Jesus is our surety that just as the lamp in the tabernacle never went out, neither will God ever stop watching over or leading His people today, and it is by means of the Holy Spirit which dwells in each of us, that He watches and leads continually.
Ex 37:25-29 and 30:1-10 describe the construction of the altar of incense.  This altar, overlaid with gold, would reside in the Holy place, directly in front of the veil leading to the Holy of Holies.  Aaron was to burn incense on this alter every morning and evening when he tended the lamp.  The incense was to burn continually.
Incense often represents prayers of the saints in the Bible (see Psalm 141:2 and Revelation 8:3-4).  I find it interesting, however, that it was the High priest, Aaron who is instructed in these passages tend this altar and keep the incense burning.  In other scriptures it is clear that other priests would tend the incense altar as well, but perhaps in this passage we are meant to see the symbolism representing the way in which not only may we approach the Father in prayer constantly, but Christ (as our high priest) is interceding on our behalf constantly before the Lord (Romans 8:34).

Article about the typology of the incense throughout the Bible:  https://www.christiancourier.com/articles/1410-old-testament-and-incense-the

Ex 38:1-7 and 27:1-8 Describes the building of the altar of burnt offering.  This largest piece of furniture in the tabernacle was positioned inside the outer court near the entrance to the tent of meeting (Ex 40:6), and it is where sacrificed animals were burnt by the priests to make atonement for themselves and for the people.

Leviticus 1 tells us that when someone wanted to offer a sacrifice they brought it to the entrance of the tabernacle and there laid their hand on its head, and then slaughtered it themselves.  Then a priest would carry the blood to the alter and burn the body as an offering to God.

Leviticus 6:12-13 says that the fire on the altar was to burn continually and never go out.
Leviticus brings more light to the purpose and use of the altar of burnt offering.  The people were called to sacrifice animals in atonement for their sin, just as Jesus would one day be sacrificed as our ultimate propitiation.  This altar, near the entrance, is where the animals, once killed by the priest or person presenting the offering, is where the blood of the animal would be offered and the priests would burn the body of the animal as a pleasing aroma to God.  In the same way Christ offered His body to appease the wrath of God for us, we must offer our bodies in service to God each day, as a spiritual sacrifice for the Lord.  Jesus, as our high priest, will bring our sacrifice before the Lord as a pleasing aroma.

Perhaps the continual burning of the fire on the altar was a reminder of the people of the continual offer of relationship and atonement that God offers.  This is certainly the nature of God’s offer to us today.
Ex 30:17-21 and 38:8 God instructed a bronze basin to be made of mirrors for the purpose of washing for the priests.  It would be placed between the altar of burnt sacrifice and the entrance to the Holy Place.  Anytime a priest was going to enter the Holy Place or approached the alter to minister, they were to wash their feet and hands to avoid judgment for uncleanliness

It is interesting that the basin was made with mirrors.  Whenever a priest washed after making a sacrifice, he would see himself in the basin.  They would know their earthly nature and recognize that the cleanness being bestowed upon them was not of them but of God.
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The position of the laver seems to be significant. If a priest wanted to approach God in the Holy Place (or even beyond that in the Most Holy Place) they first would enter the outer court and approach the altar of burnt sacrifice where they would offer a blood sacrifice, symbolizing the death of Christ as payment for their sins.  After that they would come to the bronze laver where they would wash before entering the Holy Place.  The washing with water came after the sacrifice.  We cannot be made clean to approach Jesus, but Jesus makes us clean so that we may approach God.  We come to Jesus just as we are, in need of His sacrifice, and only after receiving salvation are we made clean so that we may commune with God.  http://www.bible-history.com/tabernacle/TAB4The_Bronze_Laver.htm

Similarly, in John 13, Jesus observed the Lord’s supper with his disciples (symbolizing the sacrifice He would be for them shortly) and then immediately washed their feet with water.  He then stated that they were clean, all except one, referring to Judas.  However, Judas was among them and surely had his feet washed as well (or surely some hubbub would have arisen over it).  So, it is clear that in this washing of their feet Jesus was trying to teach something more significant than just physical cleanliness or even acts of service for one another.

Just as the priests would see their reflection in the basin when they washed, we, when we come to Christ, must acknowledge who we are without Him, and then accept who we become in Him.  Recognition of our identity is huge as believers in Christ.  He has made us clean.  We must see that and embrace it.

Question: In 1 Corinthians 7:13-14 we are told that not only is a child of God made clean, but his or her spouse and children are made clean as well, because of their status as a believer.  If the cleansing in the Old Testament signified one’s ability to now approach God and commune with Him in His presence, does this hold true now for the children of believers and even an unbelieving spouse? 
Ex 38:9-20 describes the outer court.  This large enclosed area had only one entrance at the eastern side, and otherwise was enclosed by a barrier of fine white linen.
Similar to the layout of the outer court, there is only one way to enter the kingdom of heaven.  Also, Revelation 19:8 tells us that the bride of Christ will be adorned in fine white linen.  Here we see the way in which the tabernacle itself represented the believer—the bride of Christ.
Ex 40:1-35 the tabernacle is erected, and all of its furnishings are anointed.  The Priests also are dressed for service and anointed to be a “perpetual priesthood.  And the glory of the Lord filled the temple.
When we come into God’s service, we become like this tabernacle, anointed for Holy service.  We also become dressed and consecrated like the priests, prepared for perpetual, unending service to the Lord.  The Lord fills us with His glory – for we have none of our own.
Ex 40:36-38 describes the way that God showed His presence among the people over His tabernacle.  He led them to stay or move on through the movement of a cloud over the tabernacle.
I’m not sure how much parallel we can really draw from the way that God led His children in Exodus, except that He did lead them very clearly.  He gave them no doubt regarding when it was time to move on.  I think similarly He leads His priests today.  He will be present with us in the things He calls us to, but if we find ourselves laboring in an area without His blessing and work, it may be a sign that He would have us move on to something different where He is at work.